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arXiv:1710.08814 · Physical review. B./Physical review. B

Full counting statistics of the subsystem energy for free fermions and quantum spin chains

Khadijeh Najafi, M. A. Rajabpour

Published Dec 7, 2017• 39 citations
arXivHTMLPDF

Abstract

We calculate the full counting statistics (FCS) of a subsystem energy in free fermionic systems by means of the Grassmann variables. We demonstrate that the generating function of these systems can be written as a determinant formula with respect to the Hamiltonian couplings and by using the Bell's polynomials, we derive exact formulas for the subsystem energy moments. In addition, we discuss the same quantities in the quantum $XY$ spin chain, and we demonstrate that at the critical regimes the fluctuations of the energy moments decay like a power law as we expect from the conformal field theory arguments, while in noncritical regimes, the decay is exponential. Furthermore, we discuss the full counting statistics of subsystem energy in the quantum $XX$ chain.

DOI
10.1103/physrevb.96.235109

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We calculate the full counting statistics (FCS) of a subsystem energy in free fermionic systems by means of the Grassmann variables. We demonstrate that the generating function of these systems can be written as a determinant formula with respect to the Hamiltonian couplings and by using the Bell's polynomials, we derive exact formulas for the subsystem energy moments. In addition, we discuss the same quantities in the quantum XYXYXY spin chain, and we demonstrate that at the critical regimes the fluctuations of the energy moments decay like a power law as we expect from the conformal field theory arguments, while in noncritical regimes, the decay is exponential. Furthermore, we discuss the full counting statistics of subsystem energy in the quantum XXXXXX chain.
Abstract The spin-1/2 Heisenberg chain exhibits a quantum critical regime characterized by quasi long-range magnetic order at zero temperature. We quantify the strength of quantum fluctuations in the ground state by determining the probability distributions of the components of the (staggered) subsystem magnetization. Some of these exhibit scaling and the corresponding universal scaling functions can be determined by free fermion methods and by exploiting a relation with the boundary sine-Gordon model.
We show that the computation of formation probabilities (FP's) in the configuration basis and the full counting statistics of observables in quadratic fermionic Hamiltonians are equivalent to the calculation of emptiness formation probability (EFP) in the Hamiltonian with a defect. In particular, we first show that the FP of finding a particular configuration in the ground state is equivalent to the EFP of the ground state of the quadratic Hamiltonian with a defect. Then, we show that the probability of finding a particular value for any quadratic observable is equivalent to a FP problem and ultimately leads to the calculation of EFP in the ground state of a Hamiltonian with a defect. We provide exact determinant formulas for the FP in generic quadratic fermionic Hamiltonians. For applications of our formalism we study the statistics of the number of particles and kinks. Our conclusions can be extended also to quantum spin chains, which can be mapped to free fermions via Jordan-Wigner transformation. In particular, we provide an exact solution to the problem of the transverse field XYXYXY chain with a staggered line defect. We also study the distribution of magnetization and kinks in the transverse field XYXYXY chain numerically and show how the dual nature of these quantities manifests itself in the distributions.
This thesis investigates the dynamics of quantum many-body systems in isolation, in three different setups. We start by considering a paradigmatic quantum chaotic model of magnetism, the Heisenberg antiferromagnet in 2 and 3 spatial dimensions. The focus is on obtaining approximations for the full counting statistics (FCS) of a subsystem order parameter, the staggered magnetization, both in and out of equilibrium (following quantum quenches). We employ mappings of spin variables to bosonic ones, together with self-consistent time-dependent mean-field schemes, to compute the FCS via Gaussian techniques. In particular, we introduce a formula for the FCS of any subsystem observable that is at most quadratic in the bosons. We obtain results both in the presence and absence of long-range magnetic order. We then turn to the first of two ergodicity-breaking scenarios, by considering Hilbert space fragmentation (HSF) in charge- and dipole-conserving quantum chains. We determine the critical charge density at which a "freezing" phase transition from weak to strong fragmentation takes place, for any on-site Hilbert space dimension. To characterize the degree of dynamical connectivity in the system, we introduce a novel framework that enables us to obtain analytic results for the strongly fragmented phase, and numerically explore the weakly fragmented one. We derive some critical exponents, and show that the entanglement entropy of typical eigenstates in the strongly fragmented phase obeys an area law. We also uncover the existence of rare eigenstates whose entropy appears to scale beyond any area law. Finally, we investigate dynamical correlators in the Lieb-Liniger gas, one of the simplest interacting integrable theories solvable by Bethe ansatz. We employ Markov chain Monte Carlo methods to sample matrix elements (form factors) that appear in the Lehmann representation of the finite-temperature single-particle bosonic Green’s function. In the physical regimes we consider, sampling is necessary because a full summation of the relevant form factors is computationally intractable. We benchmark our results against exact solutions available in specific limits, and find them to be very accurate.
We present analytical and numerical results regarding the magnetization full counting statistics (FCS) of a subsystem in the ground state of the Haldane-Shastry chain. Exact Pfaffian expressions are derived for the cumulant generating function, as well as any observable diagonal in the spin basis. In the limit of large systems, the scaling of the FCS is found to be in agreement with the Luttinger liquid theory. The same techniques are also applied to inhomogeneous deformations of the chain. This introduces a certain amount of disorder in the system; however we show numerically that this is not sufficient to flow to the random singlet phase, that corresponds to XXZXXZXXZ chains with uncorrelated bond disorder.
An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system.We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models.The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered.Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures.We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.
We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization, and the domain walls at zero temperature for a finite interval of the XY spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlevé V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a by-product, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both the σ^{z} and σ^{x} basis in the ground state. The analysis hinges upon asymptotic expansions of block Toeplitz determinants, for which we formulate and check numerically a different conjecture.